Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) = #496097 (order 512) · Inn(G) = C42 · Out(G) = C4×D4 |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C42 |
| commutator | C4 | C4×C16 |
| Frattini, cosocle | C2×C32 | C22 |
| socle | C2 | #1397 |
| prime radical | C22 | M64 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
37 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,3,9,4,11,15,30,5,12,16,31,18,34,39,55,6,13,17,28,19,35,40,51,21,36,41,52,42,57,59,63,7,10,14,29,20,32,37,53,22,33,38,54,43,56,58,64,23,8,24,25,44,26,46,48,45,27,47,49,60,50,61,62);(2,8,10,13)(3,14)(9,28,29,25)(11,26,32,35)(12,27,33,36)(15,37)(16,38)(17,24)(30,51,53,48)(31,52,54,49)(34,50,56,57)(39,58)(40,46)(41,47)(55,63,64,62)(59,61)Polycyclic presentation:
2^8|a^2=c,b^2=d,c^2=e,e^2=f,f^2=g,g^2=h,b^a=bg,d^a=dh,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #94506, #94509, #94613, … |
| maximal normal subgroup of | #94506, #94509, #94613, … |
| maximal quotient of | #94505, #94508, #94756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 499);